Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Definition 4.6.2.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that $F$ is fully faithful if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map of morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ is a homotopy equivalence of Kan complexes.